I read from a Forier Analysis book that the space $L^p(\Omega)$ is a quasi-Banach space for $p\in (0,1)$. The book mentions that $||\cdot||_p$ is a quasi-norm, which means that $$ ||f+g||_p \le C (||f||_p + ||g||_p) $$ for some $C\ge 1$ (of course, one can easily prove that $C=2^{\frac 1p-1}$ works here). However, it was not mentioned explicitly what a quasi-Banach space means.
What is the definition of a quasi-Banach space and how do we prove that $L^p(\Omega)$ is quasi-complete for $p\in(0,1)$?
I can guess what the phrase ought to mean but I am not completely sure.
Let $X$ be a vector space together with a quasi-norm $\|\cdot\|_{quasi}$. One can show (Aoki-Rolewicz Theorem) that there exists $r \in (0,1]$ such that $d(x,y) := \|x-y\|^r_{quasi}$ is a metric. In particular, we can ask if $(X,d)$ is complete and then we say that $(X,d)$ is a quasi-Banach space.
In our special case, one can define a metric on $L^p(\Omega,\mathcal{A},\mu)$, $0 < p <1$, via $$d(f,g):= \int |f-g|^p \, d \mu.$$ This metric is invariant under translations (i.e. $d(f-h,g-h) = d(f,g)$), but not a norm. In particular, $L^p$ is a topological vector space. Using the same arguments as in the proof of the completness of $L^p$ with $p \geq 1$, one can show that $L^p$ is complete with respect to the metric $d$.
The interesting part here is that it is possible to show that $(L^p)^* = \{0\}$ for measures $\mu$ without atoms. Thus, in view of Banach's extension theorems, there is no norm on $L^p$ inducing the above-mentioned topology. In particular, $L^p$ is for $0 <p <1 $ not a locally convex topological vector space.